NASA Technical Reports Server (NTRS) 19650018212: Determination of plasma temperature and electron density distributions using millimeter waves

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DETERMINATION OF
PLASMA TEMPERATURE AND
ELECTRON DENSITY DISTRIBUTIONS
USING MILLIMETER WAVES

by William F. Leonard

Langley Research Center
Langley Station, Hampton, Va.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • JULY 1965

DETERMINATION OF PLASMA TEMPERATURE AND

ELECTRON DENSITY DISTRIBUTIONS

USING MILLIMETER WAVES*

By William F. Leonard
Langley Research Center

SUMMARY

y7&3

A method for obtaining good spatial resolution in the measurement of elec-
tron density and temperature variations in a thermal plasma of cylindrical
cross section using millimeter waves is described. The technique, which is an
application of the Abel inversion technique, involves the division of a plasma
into concentric zones and evaluation of the attenuation constant in each zone
from measured attenuation losses. Results of measurements made on a cyanogen
oxygen flame at 61.2 Gc are given, and correlation of the peak temperature
(4470° K) at the center of the flame with spectroscopic measurements is shown.

INTRODUCTION

As a result of high -temperature plasma research, various techniques have
been developed to measure the electromagnetic properties of an ionized gas.

These methods fall into the following categories: metallic current probes,

electron beam probes, optical measurements of emission line broadening and of
emission spectra, and schemes employing microwave interactions with the medium.
Probe techniques are often not applicable due to a lack of suitable theories
for interpreting the data they supply, and optical measurements may be limited
because of low spectral line intensity. Therefore, efforts have been extended
toward the use of microwave diagnostics (refs. 1, 2, and 3) to measure the
electromagnetic properties of a plasma (refs. 4, 5> and 6). Of particular
interest is the determination of plasma temperature and electron density. Some
previous studies of plasma temperature and electron density (refs. 7 and 8) had,
in general, low spatial resolution and resulted in average values for the elec-
tron density and temperature.

The purpose of this report is to describe a high-resolution millimeter-
wave survey technique for studying electron density and temperature distribu-
tions in nonreflecting stratified cylindrical plasmas. This is done by first

^Presented at the Millimeter and Submillimeter Conference, Orlando,

Florida, January 7-10, 1963 , sponsored by The Institute of Electrical and
Electronics Engineers.

describing the conditions necessary for a plasma to be nonreflecting, then
applying a survey method based on the Abel inversion technique to an assumed
cylindrical model for a plasma. Experimental results obtained on a cyanogen
oxygen flame are then presented and compared with spectroscopic measurements
of the peak temperature at the center of the flame.

SYMBOLS

area coefficient

B degeneracy factor in Saha's equation

K equilibrium constant for dissociation or ionization

l path length, cm

N e electron density, cm-3

r^ radius vector

T absolute temperature, °K

ionization potential, eV

xj component of radius vector perpendicular to transmission path

a attenuation constant, dB/cm

p phase constant, rad/ cm

Tp power reflection coefficient

7 propagation constant

6 attenuation , dB

A wavelength, mm

v collision frequency, sec~l

Subscripts:

i summation index

j,k zone indices for area coefficients

n summation

2

o

free space

p plasma

SURVEY METHOD

The propagation of electromagnetic EM waves in a plasma is described in
terms of a complex propagation constant y = a + jfj which is a function of the
electron density N e , temperature and collision frequency v of the plasma.
(See ref. 4.) In general, to determine the electron density, temperature, and
collision frequency of a plasma both the attenuation constant a and the phase
constant 3 must be known. However, under certain conditions (figs. 10 and 15

of ref. 5)> if either

EM frequency

> 10-L0 and

N,

1/2

EM frequency

^ lO 5

or

N e l/2

EM frequency

S 3* 16 x 10 “2 then the phase shift in the plasma is equal to that

of free space and the power reflection coefficient Tp is less than 10

-4

Thus, for a "nonreflecting" plasma the experimental quantity of interest is the
attenuation constant. An obvious means to insure these above criteria is to
increase the EM frequency; this has the added benefit of obtaining good spa-
tial resolution in addition to simplifying the experimental measurements. That
is, for systems operating at EM frequencies greater than 60 Gc, the physical
dimensions of the horn antennas are less than 0.3 inch, and Buser and Buser
(ref. 9 ) have experimentally shown that the beam is well collimated and is
approximately equal to the width of the horns. Thus, if the ratio of plasma
diameter (assuming a cylindrical plasma) to the width of the antenna receiving
aperture is large, good spatial resolution can be obtained.

The effect of temperature on the EM properties of a plasma is determined
through the use of known gas equilibrium constants and Saha's equation. Saha's
equation relates the electron density to the temperature in an ionized gas for
a given ionization potential Vp and total gas pressure. Saha's equation is

log K p =

23070 Vp , .

+ 2.5 log T + log B - 6.491

4.573 T

( 1 )

where K p is the equilibrium constant for dissociation or ionization of the

various constituents of the plasma and B is the degeneracy factor. (See
ref. 8 .) If the known equilibrium constants and those calculated from equa-
tion (l) are used, a composition calculation for neutral particles and elec-
trons can be made.

3

r

<

If it is assumed that r p = 0 (or 0 = p 0 ), one further requirement

becomes necessary to determine N e and T from attenuation measurements; that

is, the value of the collision frequency must be known. This is necessary

because the slope of the curve relating phase shift to electron density

(fig. 10 of ref. 5) is zero regardless of the value of the collision frequency.
However, for many plasmas the collision frequency may be computed from kinetic-
theory relations. (See eq. ( 4 ) of ref. 8 .) Thus, with a measured attenuation
constant, a calculated collision frequency, and a knowledge of gas constituents,
the electron density and temperature can be determined.

In order to arrive at a reasonable survey method based on these criteria,
the following plasma conditions are assumed;

1. The plasma is nonreflecting at the EM frequency used.

2. The plasma is cylindrical, having radial variations only.

3. The ratio of plasma diameter to width of antenna receiving aperture is
large .

4 . The ionization potential, total gas pressure, and collision frequency
of the plasma are known.

Once these plasma conditions are assumed, the electron density and temperature
variations in a plasma can be determined by measuring the insertion loss experi-
enced as the test plasma traverses between two microwave horns normal to the
direction of propagation.

Since theory is based on
attenuation per unit path
length, a model for evaluating
effective path lengths will be
helpful in converting from
measured attenuation in dB to
attenuation per unit path
length in dB/cm. Figure 1,
which represents the model
used to evaluate laboratory
tests, shows a cylindrical
plasma divided into five con-
centric zones of constant
attenuation per unit path
length with a width equal to
that of the receiving horn
aperture. The number of
zones depends on the ratio of
the plasma diameter to the
width of antenna receiving
aperture. The effective path

4

V

length in a zone is found by dividing the area of the strips in that zone by
the width. The area of a strip in a particular zone is obtained from a table
of area coefficients A]^j for given values of r k and x^ (ref. 10).

Once the effective path lengths and measured attenuation losses are known
for each strip , the attenuation constant for each zone can be determined from
the following expression:

n n-1

where the subscript n designates the zone starting with n = 1 for the outer
zone and 5^ is the attenuation in decibels in a zone for a particular strip.

If the attenuation constants thus obtained are used, the electron density and
temperature in each zone are found from theoretical plots of attenuation per
unit length versus electron density and electron density versus temperature for
the plasma being surveyed.

TEST APPARATUS

The test plasma used to
evaluate the survey technique
was a stoichiometric cyanogen
oxygen flame which forms a
3 -inch-diameter subsonic jet at
atmospheric pressure. The cal-
culated collision frequency for
this flame is 6 x lO^O sec - l
(ref. 8), and computation of
the equilibrium plasma char-
acteristics for a stoichiometric
equilibrium combustion of cyano-
gen and oxygen gives a plot of
temperature versus electron den-
sity as shown in figure 2.

Examination of the electro-
magnetic properties of a plasma
at a frequency of 6l.2 Gc (fre-
quency used in tests) and for a
collision frequency of
6 x 10 10 sec - -*- yields the
following:

Figure 2.- Variation of electron density with temperature for the cyanogen
oxygen flame.

5

(1) The cyanogen oxygen flame
is nonreflecting for electron den-
sities less than 10-*-5 cm" 5 . (see
fig. 10 of ref. 5.)

( 2 ) A plot of attenuation con-
stant versus electron density is as
shown in figure 3-

By combining figures 2 and 3, a
useful plot of temperature versus
attenuation constant can be made
and is shown in figure 4 .

Therefore, the cyanogen oxygen
flame meets the necessary require-
ments outlined and a temperature
distribution can be made by meas-
uring transmission loss as a func-
tion of flame diameter.

Figure 5 is a block diagram of
the millimeter-wave apparatus. The
signal source is a 60 to 70 Gc
backward wave oscillator which
feeds into a 3 -dB coupler for

Figure 3.- Dependence of attenuation constant on electron density,
v = 6 x 10 IO sec'4 X = 4.9 mm.

Figure 4.- Dependence of attenuation constant on temperature, u = 6 x 10 10 sec“4

X = 4.9 mm.

Figure 5.- Schematic diagram of test apparatus.

monitoring power output levels.
Frequency of operation is checked
by measuring the wavelength with a
slotted line. The antennas are
15 -dB nominal gain horns with an
aperture width of 0.3 inch. An
oscillograph recorder is used to
record the power monitor signal,
flame position, and transmitted
signal. Figure 6 is a photograph
of the test facility.

RESULTS

Attenuation measurements were
made on a stoichiometric cyanogen
oxygen flame at a frequency of
61.2 Gc. The ratio of flame diame-
ter to width of receiving horn
antenna was 10:1; thus, the flame
was divided into five concentric
zones each having a width of
0.3 inch (the width of the
receiving horn aperture).

L-65-16

Figure 6.- Photograph of test facility.

The measured attenuation experienced as the flame traversed between the
horns is shown in figure 7* Application of these data to the assumed plasma
model yields the attenuation constant for each zone. The resulting electron
density and temperature distributions, from figures 3 an d- ^ anc ^ values obtained
for the attenuation constants, for the cyanogen oxygen flame are shown in
figure 8.

Although some fluctuations were present in the flame, the survey technique
gave a distribution which follows a bell-shaped curve with a maximum temperature

7

Figure 7.- Transverse survey of cyanogen oxygen flame at 61.2 Gc.

at the center of 4470° K
and a minimum of 4l45° K
at the edge. The peak
temperature at the center
of the flame has been cor-
related with unpublished
spectrographic data,
obtained at the Langley
Research Center, based on
the rotational structure
of the vibrational rota-
tional CN band. (See
ref. 11 for a description
of this technique.) The
spectrographic measure-
ments gave a peak value of
4500° K at the center of
the flame which is within
1 percent of the millimeter
wave value.

The flare of the elec-
tron density and tempera-
ture distributions in fig-
ure 8 results from the
assumption that the atten-
uation coefficient a is
a constant in each zone.
This effect becomes negli-
gible in the calculation
of a (eq. (2)) toward
the center of the flame
because the total attenua-
tion contributed by the
outer zone is small.

An advantage of this
method for studying the
temperature variations in
the cyanogen oxygen flame
is that the results are
not very sensitive to the
This can be shown by comparing

versus N e ~' “ for various values of v (fig. 1 of ref. 5) and
figure 2 for T versus N e . If the collision frequency is in error by an order
of magnitude, the error in the temperature values determined from attenuation
measurements at 61.2 Gc will be less than 12 percent. The shape of the tempera-
ture distribution curve does not change with collision frequency.

Figure 8.- Electron density and temperature distributions in a cyanogen oxygen flame.

accuracy of the calculated collision frequency
a plot of a versus N e ^' ^

8

CONCLUDING REMARKS

Electron density and temperature distribution of a nonreflecting cyanogen
oxygen flame have been determined by using a millimeter-wave survey scheme.

The distributions follow the expected bell-shaped curve and the peak tempera-
ture at the center of the flame agrees within 1 percent of spectroscopic
measurements .

Langley Research Center,

National Aeronautics and Space Administration,

Langley Station, Hampton, Va., January 21 , 1965.

9

REFERENCES

1. Drummond, James E.: Plasma Physics. McGraw Hill Book Co., Inc., 1961.

2. Wharton, Charles B.: Microwave Diagnostics for High -Temperature Plasmas.

UCRL-4836 (Contract No. W-7405-eng-48) , Univ. of California, Mar. 1957*

3. Bunn, Harlin L.: Microwave Diagnostic Systems and Techniques for Use in

Controlled Fusion Research. IRE, Trans. Instr., vol. 1-11, no. 1, June

1962, pp. 3-10.

4. Bacbynski, M. P.; Johnston, T. W. ; and Shkarofsky, I. P.: Electromagnetic

Properties of High-Temperature Air. Proc. IRE, vol. 48, no. 3> Mar. i960,
pp. 3^7-356.

5. Balwanz, W. W.: Interaction Between Electromagnetic Waves and Flames.

Pt. 6 - Theoretical Plots of Absorption, Phase Shift, and Reflection.

NRL Rept. 5388, U.S. Naval Res. Lab., Sept. 23, 1959-

6. Buchsbaum, S.: On the Interaction of Microwave Radiations With a Plasma.

Semi-Annual Progress Report. ZPH-013 (Contract AF 33(6l6) -3283) , Convair-
San Diego, Apr. 1, 1958, pp. 77-89*

7- Rudlin, Leonard: Preliminary Results of a Determination of Temperatures of
Flames by Means of K-Band Microwave Attenuation. NACA RM E51G20, 1951.

8. Huber, Paul W . ; and Gooderum, Paul B. (With appendix A by Theo E. Sims and
Duncan E. Mclver, Jr., and appendix B by Joseph Burlock and William L.
Grantham): Experiments With Plasmas Produced by Potassium-Seeded

Cyanogen Oxygen Flames for Study of Radio Transmission at Simulated
Reentry Vehicle Plasma Conditions. NASA IN D-627, 1961.

9* Buser, R.; and Buser, W. : Determination of Plasma Properties by Free-Space

Microwave Techniques. J. Appl. Phys., vol. 33 j no. 7 , July 1962,

pp. 2275-2282.

10. Pearce, William J.: Calculation of the Radial Distribution of Photon

Emitters in Symmetric Sources. Conference on Extremely High Tempera-
tures, Heinz Fischer and Lawrence C. Mansur, eds., John Wiley &. Sons,

Inc., c. 1958, pp. 123-134.

11. Greenshields, David H. : Spectrometric Measurements of Gas Temperatures in

Arc-Heated Jets and Tunnels. NASA TN D-I960, 1963.

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NASA-Langley, 1965 L-4276